Issue 
Eur. Phys. J. Appl. Phys.
Volume 88, Number 2, November 2019



Article Number  20904  
Number of page(s)  9  
Section  Physics of Energy Transfer, Conversion and Storage  
DOI  https://doi.org/10.1051/epjap/2019190284  
Published online  05 February 2020 
https://doi.org/10.1051/epjap/2019190284
Regular Article
The molar mass of a new enriched silicon crystal: maintaining the realization and dissemination of the kilogram and mole in the new SI
PhysikalischTechnische Bundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Germany
^{*} email: axel.pramann@ptb.de
Received:
13
September
2019
Received in final form:
20
December
2019
Accepted:
24
December
2019
Published online: 5 February 2020
The local distribution of the isotopic composition and molar mass M of a new silicon crystal (Si2824Pr11) highly enriched in the ^{28}Si isotope is reported, with focus on the experimental methods as well as on the associated uncertainties. The crystal was used in 2018 for the production of two additional silicon spheres for the realization and verification of the Avogadro constant N_{A} using the “Xraycrystaldensity (XRCD) method” which is a primary method for the dissemination of the revised SI units mole and kilogram. 17 subsamples have been investigated (from five different axial and in several radial positions) by isotope ratio mass spectrometry using a multicollectorinductively coupled plasma mass spectrometer (MCICPMS). The average molar mass of the crystal is M = 27.976 933 787(77) g/mol with a relative combined uncertainty u_{c,rel}(M) = 2.7 × 10^{−9}. The mean amountofsubstance fraction of ^{28}Si is x(^{28}Si) = 0.999 993 104 (66) mol/mol indicating that this crystal has the highest enrichment in this isotope which has ever been used for the determination of N_{A}. No local variations in M and x(^{i}Si) (i = 28, 29, and 30) could be identified due to material properties. The results are compared with those from two previous enriched crystals.
© EDP Sciences, 2020
1 Introduction
The International System of Units (SI) was revised and the revision came into effect from May 20, 2019. This has been decided by the General Conference on Weights and Measures (CGPM) in November 2018 at its 26th meeting [1,2]. From that date, the SI base units are defined via fixed fundamental constants [3,4]. The SI unit of the mass, the kilogram, is now defined via the Planck constant h, and the mole, the SI unit of the amount of substance, is defined via the Avogadro constant N_{A}. The best experimental methods to date to realize and disseminate these fundamental constants are the Kibble balance (watt balance) for the determination of h by comparing electrical power with mechanical power [5] and the Xraycrystaldensity (XRCD) method − using silicon spheres which are chemically ultrapure and highly enriched in ^{28}Si for the determination of the Avogadro constant N_{A} [6–8]. In Metrology, the most appropriate methods are indicated as “primary methods” yielding the respective quantities with lowest associated uncertainties.
By counting silicon atoms (XRCD method), N_{A} is measured using a sphere of enriched silicon with an approximate mass of 1 kg by(1)(with 8 atoms in the unit cell, the sphere volume V, its mass m, the lattice parameter a, and the molar mass M). N denotes the number of Si atoms in the sphere and n the respective amount of substance [8]. After the revision of the SI, N_{A} is fixed numerically and its associated measurement uncertainty is set to zero by definition. When a new crystal material or sphere in the future will be available, all parameters in the centre of equation (1) have to be determined with lowest associated uncertainties using the XRCD method. The common route used on that level for the dissemination of the SI units kilogram and mol will be as follows: the characterized Si sphere can be used as a primary standard for the mass m (with an uncertainty) according to(2)(with the Rydberg constant R_{∞}, the speed of light in vacuum c, the fine structure constant α, the Planck constant h, the relative atomic mass of the electron A_{r}(e), the relative atomic masses A_{r}(^{i}Si) of the silicon isotopes, the surface layer mass m_{sur}, and the mass of crystal point defects m_{def}). The amount of substance n is disseminated on the highest level via(3)(with the molar mass constant M_{u}). Moreover, equation (3) shows the relation between h and N_{A}. The production of several highly enriched silicon spheres (from different crystal ingots, each from different production processes) thus serves as a way to generate a pool of primary standards for the mass and the amount of substance used for their dissemination with lowest uncertainties, accessible for science and industry.
In this article, we present the measurement of the molar mass M and isotopic composition x(^{i}Si) of a new highly enriched silicon crystal (Si2824Pr11), finished after the deadline for the data used for the revision of the SI. Thus, the data are important for the validation, verification and dissemination of m and n. The new crystal shown in Figures 1 and 2 has been produced during a jointproject (“kilogram2” or “kg2”) of PTB and Russian institutes and companies [9,10]. We report on the measurement of 17 discrete silicon samples (each with a gross weight of approximately 500 mg) from five different axial positions in the original ingot bracketing the location of the two spheres intended for the XRCD measurements. On each axial position, up to six adjacent radially arranged samples were measured. This could enable an assessment of the variation of M and x(^{i}Si) as a function of the origin in the crystal.
The data evaluation is based on an uncertainty analysis according to the “Guide to the Expression of Uncertainty in Measurement” (GUM) [11]. The results are compared to the two other available enriched crystals Si2810Pr11 and Si2823Pr11 which were characterized previously [12,13].
At present, six spheres from three different crystals highly enriched in ^{28}Si have been completely characterized using the XRCD method (two spheres from each crystal). Starting in 2007, the first two enriched spheres (AVO28S5 and AVO28S8) from the crystal Si2810Pr11 were available. In the “kilogram2” project, two spheres (Si28kg01a and Si28kg01b from the crystal Si2823Pr11, available in 2015) and another two spheres (Si28kg02a and Si28kg02b from the crystal Si2824Pr11, available in 2016, this work) have been produced. In the near future, six more enriched spheres (“kilogram3” project) will be available, finally yielding a set of twelve enriched silicon spheres. Theoretically, the uncertainty associated with the molar mass should be smaller with increasing enrichment in the ^{28}Si isotope as discussed in [13]. However, a higher enrichment might complicate the measurement and thus slightly increase the uncertainty as is shown in this work, meaning from the point of view of the molar mass determination there is an optimum enrichment.
Fig. 1 Photograph of the final single crystalline silicon crystal ingot Si2824Pr11 (length: 52 cm, mass: 5575 g) used for the spheres Si28kg02a and Si28kg02b. 
Fig. 2 Final silicon crystal ingot (Si2824Pr11). Left: in the float zone apparatus at the Institute of Crystal Growth (IKZ), Berlin, Germany; right: main parts already cut (the large parts are reserved for the two spheres). 
2 Molar mass via isotope ratio measurements
Isotope ratio measurements via high resolution mass spectrometry offer the best access to analyte concentrations (mass fractions) or isotope distributions via amountofsubstance fractions x or even to a molar mass M. In case of the enriched silicon material, all three natural isotopes ^{28}Si, ^{29}Si, and ^{30}Si have to be taken into account, although the two latter are by six and seven orders of magnitude less abundant in the crystal. To overcome the technical problems of absolute measurements of these isotopes in trace amounts and to obtain sufficiently low uncertainties, a special modified method of the classical isotope dilution mass spectrometry (IDMS) technique has been developed [14,15]. This technique denoted as “virtual element” (VEIDMS) is now routinely applied to determine the molar mass of enriched silicon yielding relative uncertainties u_{rel}(M) < 5 × 10^{−9}. In combination with a highresolution MCICPMS, this method guarantees most precise and accurate results of both the isotopic composition and the molar mass and has been applied and validated by various national metrology institutes (NMIs) [16–19]. The measured “intensity” ratios are biased by nature due to space charge effects in the plasma ion source. Therefore, the measured ratios have to be corrected by calibration factors (K factors). Parallel to the VEIDMS method, an analytical closed form approach has been developed for the absolute determination of the respective K factors which rendered the molar mass measurement of enriched silicon a primary method [12,20]. For a better understanding, the VEIDMS principle is briefly described.
To avoid the standard measurements of isotope ratios e.g. ^{30}Si/^{28}Si associated with very large uncertainties, in the VEIDMS method almost only the ratios R = n(^{30}Si)/n(^{29}Si) need to be measured. The silicon material is handled as consisting theoretically of the isotopes ^{29}Si and ^{30}Si only (“impurities”) in the matrix of the most abundant ^{28}Si. The enriched sample material (x) is blended with a silicon material enriched in ^{30}Si (y, spike). The ratio ^{29}Si/^{30}Si has to be measured in the sample, spike, and blend. Additionally, the two masses m_{yx} and m_{x} (solid spike material y and sample material x in the blend bx) have to be determined. The molar mass M is calculated via equation (4) (with R_{j}_{,2} = x_{j}(^{30}Si)/x_{j}(^{29}Si) and R_{j}_{,3 }= x_{j}(^{28}Si)/x_{j}(^{29}Si)) (4)
The M(^{i}Si) in equation (4) denotes the respective molar masses derived from the respective atomic masses given in [21]. Using a characterized material (silicon with natural isotopic composition: material w) only the calibration factor K_{2} had to be calculated. This was performed by the measurement of R_{w,2}^{meas} during an actual measurement using the known “true” isotope ratio R_{w},_{2}^{true}.
3 Experimental
The new crystal material analysed in this work has been manufactured in Russia and finally in Berlin [9,10]. In order to get information about possible variations of the molar mass and isotopic composition, samples (of an almost cubic shape) with an approximate mass of 500 mg each had to be cut from the original ingot (Figs. 3 and 4). The apparent large mass of a single sample is necessary due to the low abundance of the isotopes ^{29}Si and ^{30}Si. The ability of detecting the respective isotope ratios with sufficiently small uncertainties (<2%) requires total silicon mass fractions w(Si) > 4000 μg/g from one sample, which is comparably high when operating ICPMS instruments. Two samples (L.1.1 and L.1.2) were taken from the tip of the ingot, four samples (N.2.1–N.2.4) were cut right before the origin of first sphere. The samples from part S (S.5.1–S.5.6) bracket the second sphere together with the samples V.1.1.1V.1.1.5. and V.1.2.1 (which is on top of V.1.1.1). During the measurements the samples V.1.1.3 and V.1.1.4 showed very significant offsets from the other results due to possible contamination with natural silicon or even systematic errors during the sample preparation. Therefore, these two samples were not considered in this investigation. A single sample W.2.1 was cut from the end of the crystal (the socalled Czochralskiregion).
The processes of sample preparation and mass spectrometric measurements have been described in detail elsewhere [12,13,22]. For a better understanding, only the main procedures are mentioned. All silicon samples were treated in exactly the same way to enable the possible identification of differences in the isotopic composition and molar mass due to the origin of the sample. All samples were cleaned and the oxide layer was removed by an etching process. After exact weighing, the samples were dissolved in aqueous tetramethylammonium hydroxide (TMAH, mass fraction of the final sample solution after dilution: w(TMAH) = 0.0006 g/g). The concentrations (mass fractions) of the solutions ready for the mass spectrometric measurements were: 4000 μg/g ≤ w(x) ≤ 5000 μg/g, 2500 μg/g ≤ w(bx) ≤ 3500 μg/g, and w(w) = 4 μg/g.
The isotopic composition and molar mass measurements of the silicon crystal samples have been performed using a high resolution multicollectorinductively coupled plasma mass spectrometer (MCICPMS) Neptune™ (Thermo Fisher Scientific GmbH, Bremen, Germany) [22,23]. The operating conditions and parameters are given in Table 1.
Figure 5 shows the plasma of the ICPMS source via a new bonnet made of sapphire for a better reduction of a possible sample contamination with natural silicon from the ion source. Additional advantages of the sapphire bonnet are the increased mechanical stability compared to a boron nitride bonnet and the fact that it is transparent simplifies the monitoring of the plasma.
The mass spectrometric measurements were performed identical for each sample. Three different “samples” have to be measured in a sequence: four times in the unspiked sample x (enriched silicon) the intensity ratios (in volt) of the isotopes ^{29}Si/^{30}Si are measured. Prior to each sample solution, a blank solution containing aqueous TMAH (w(TMAH) = 0.0006 g/g) was measured in exactly the same way as the sample. These data were subsequently subtracted from the sample data to correct for blank contamination effects. Subsequently, the blend bx followed by the natural silicon solutions w was measured in the same way. K factors for mass bias correction were determined by measuring the isotope ratios R_{w,2} = I_{w}(^{30}Si)/I_{w}(^{29}Si) in the “calibration” solution (natural silicon w, material name: WASO04) at the end of the IDMSsequence to avoid a cross contamination of the enriched solutions with natural silicon.
The K factor was determined by the ratio of the correct (“true”) isotope ratios determined in [12] and the measured isotope ratios R_{w,2} = I_{w}(^{30}Si)/I_{w}(^{29}Si) in the solution w of each sequence.
Fig. 3 Schematic cross section of the silicon crystal Si2824Pr11. The samples measured were taken from parts L, N, S, V, and W. The regions N, S, and V are bracketing the two spheres in the segments P and T. Each sample (cubic shape) has an approximate mass of 500 mg. 
Fig. 4 Cross section of the disc of part V. In the lower centre region, five adjacent samples V1.1.1–V1.1.5 are indicated (compare Fig. 3). 
Operating conditions of the MCICPMS with components almost siliconfree (PFA: perfluoroalkoxy alkane, PEEK: polyether ether ketone).
Fig. 5 Inductivelycoupled plasma visible via a new sapphire bonnet type with reduced content of silicon with natural isotopic composition. 
4 Results and discussion
The samples schematically indicated in Figure 3 were measured under as equal as possible conditions. The aim was to determine the respective molar masses M and the isotopic compositions expressed in their amountofsubstance fractions x( ^{i} Si) (i = 28, 29, 30) throughout the original silicon crystal ingot Si2824Pr11 in a most comparable way. Similar studies have been conducted on the crystals Si2810Pr11 (“AVO28”) [12] and Si2823Pr11 [13]. The present crystal however, has the highest enrichment in ^{28}Si and therefore, at a first glance, a respectively lowered uncertainty associated with M was expected. But due to the increased difficulties with the measurement of the extremely low ^{30}Si signal an increased uncertainty was also possible. Each sample is usually measured six times (six sequences). In contrast to previous studies, more axial positions were chosen to have a more significant local difference of the sample origin. 17 samples have been measured at PTB, resulting in an average molar mass M = 27.976 933 787(77) g/mol with a relative combined uncertainty u_{c,rel}(M) = 2.7 × 10^{−9}.
This result already covers the scattering of the averaged results of the individual samples as shown in Figure 6.
As can be clearly seen, an additional uncertainty contribution from the scattering of the values must be added in contrast to previous studies (included in the relative u_{c,rel}(M) = 2.7 × 10^{−9} (k =1; indicated by dashed lines in Fig. 6). The relative uncertainty without the scattering contribution is only u_{c,rel}(M) = 1.2 × 10^{−9} (k =1; indicated by dotted black lines in Fig. 6).
The scattering effect has a contribution of 56% to the overall uncertainty associated with M.
Note that this plot shows only a twodimensional relation. Therefore, the uncertainty of the molar mass of the crystal Si2824Pr11 is ranging between the two previously measured crystals: u_{rel}(M, Si2810Pr11) =4.4 × 10^{−9}, and u_{rel}(M, Si2823Pr11) = 1.4 × 10^{−9} [12,13]. Usually, it is theoretically expected that the higher the enrichment x(^{28}Si), the smaller the respective uncertainty associated with M. For the current crystal Si2824Pr11 x(^{28}Si) = 0.999 993 104(66) mol/mol has been determined. This is the highest enrichment of a silicon crystal intended for the use by the XRCD method, so far. The individual combined uncertainties are to some extent <10^{−9}. However, the significant scattering of the results of the different samples had to be taken into account.
Figure 7 displays threedimensional molar mass plots for a more realistic picture of the origin of the samples (given by their axial and radial positions in mm) and their respective molar masses. In parts a and b of Figure 7, the average molar mass values of the individual samples (with uncertainties) are plotted (which is a more descriptive but less precise view than the plot in Fig. 6). However, the detailed sample indication is omitted for more clarity (b is a tilted version of a). Part c of Figure 7 is a top view on the axis origins (radial and axial) of the different samples. Axial and radial positions of the samples are given in Table 2 (part W: not shown).
To check the data for the presence of any kind of inhomogeneity, the concept of degrees of equivalence d_{i} (a common consistency check) was applied. For the successful application by the XRCD method it can be shown that the individual molar masses are consistent with the respective average molar mass within their limits of uncertainty. Equation (5) defines the d_{i} as the difference of the N =17 individual values M_{i} and the respective average M:(5)
The corresponding uncertainty is given by(6)
The d_{i} were calculated using their individual uncertainties only (not including an uncertainty contribution due to the scattering). When an individual d_{i} is smaller than its respective uncertainty, the corresponding value is consistent with the overall average.
This can be visualized by the d_{i} , and their associated expanded uncertainties U(d_{i} ) with k = 2, shown in Figure 8.
In Figure 8, five samples (L1.1; S.5.3; V.1.1.2, V.1.1.5, W.2.1 very slightly) do not cover the zero line as a criterion for consistency. This is subsequently taken into account by adding the type A uncertainty of M calculated from the 17 individual results.
The combined uncertainty of M was then calculated using equation (7) (7)with the standard deviation s based on [24–26]. Although the scattering of the values in Figure 6 is significant, the DoE analysis shown in Figure 8 clearly suggests a consistent and homogeneous property of all samples. Because of the comparably extreme enrichment in ^{28}Si, the slightly different measurement conditions show an impact on the results indicated by the scattering. As an example, the six measurements (sequences) of sample L.1.1 are displayed in Figure 9.
From Figure 9 it can be clearly deduced that scattering between different measurements appears even for an individual sample. This is not due to inhomogeneities, but it is caused by different (although tiny) experimental variations which cannot be specified. The same holds true for the comparison of the different samples which scatter in the same order of magnitude as in a single sample shown in Figure 9. For this reason, a material dependent inhomogeneity cannot be confirmed. In contrast, a homogeneous behaviour of the molar mass according to the sample origin is more plausible when considering the DoE analysis shown in Figure 8 (where the individual uncertainties without the additional contribution of the scattering were used).
As an example, a representative uncertainty budget of a single measurement of the molar mass M of sample S.5.6 using equation (4) as a model equation is given in Table 3. The uncertainty calculation using the GUM Workbench Pro™ software (version 2.4.1 392; Metrodata GmbH, Germany) has been done according to the GUM [11].
The largest uncertainty contribution to the uncertainty of M is , the measured intensity ratio ^{29}Si/^{30}Si in the blend bx with a contribution of 53%. The next notable contribution originates from R_{w,2}, the corrected isotope ratio ^{29}Si/^{30}Si in the natural “calibration” solution w. The measured intensity ratio (^{29}Si/^{30}Si) in the sample contributes only another 5%. The combined relative uncertainty (k = 1) of this very sample has been determined to u_{rel}(M) = 8.7 × 10^{−10}.
Table 4 summarizes the molar masses M and amountofsubstance fractions x(^{i}Si) obtained from measurement campaigns during the last years.
In the current study, the Si crystal has the highest enrichment in ^{28}Si with x(^{28}Si) > 0.999 99 mol/mol. As predicted in a previous study [13], the higher x(^{28}Si) the smaller the associated uncertainty and the respective uncertainty of M. However, the average uncertainty associated with M in this study (Si2824Pr11) increases, compared to the crystal measured in the previous study (Si2823Pr11) due to additional uncertainty contributions from data scattering. Since the average uncertainties associated with the molar mass are <5 × 10^{−9}, all six spheres produced from the three crystals are suitable for the use in the XRCD method and moreover suitable as primary reference standards for the dissemination of the kilogram and the mole in the future.
Fig. 6 Average values of molar masses of the 17 individual crystal samples of the new crystal Si2824Pr11 measured in the current study. Combined uncertainties (k = 1) are indicated by error bars. The arithmetic mean is M = 27.976 933 787(77) g/mol (indicated by the solid line). The relative combined uncertainty associated with the average molar mass is u_{c,rel}(M) = 2.7 × 10^{−9} (including the contribution of scattering). Upper and lower limits of this uncertainty are shown by red dashed lines. The dotted black lines indicate the upper and lower uncertainty limits without the contribution of scattering. 
Fig. 7 Threedimensional molar mass plots showing the axial and radial origins of the samples together with the respective molar masses. The plots serve as a rough indication of the sample origin in the initial crystal ingot (Fig. 1). Detailed molar mass distributions are shown in Figure 6. 
Exact positions of the silicon samples in the crystal and respective molar masses.
Fig. 8 Degrees of equivalence d_{i} of the results of the average values of the molar mass of the 17 crystal samples. Error bars denote the expanded uncertainties (k = 2) associated with the d_{i} . Note that data encompassing the zero line with their uncertainties are consistent with the average molar mass. This uncertainty still does not cover the scattering described in the text. 
Fig. 9 Individual measurements (sequences) of the molar mass of the crystal sample Si2824Pr11L.1.1 with the individual combined uncertainty (k =1). The solid line represents the average molar mass of sample L.1.1 with upper and lower limits of the average uncertainty (dashed lines). 
Sample S.5.6: representative uncertainty budget of a single measurement of M.
Arithmetic mean values of M, x(^{28}Si), x(^{29}Si), and x(^{30}Si) of the different available Si crystals highly enriched in ^{28}Si. Numbers in brackets denote uncertainties in the last digits (k = 1).
5 Conclusion
We report on the examination of the distribution of the molar mass M and the isotopic composition expressed by the respective amountofsubstance fractions x(^{i}Si) measured in a new silicon crystal highly enriched in ^{28}Si. This is the third crystal ever available to disseminate the mole and the kilogram via the XRCD method providing two additional silicon spheres usable as primary reference standards for this purpose. The new crystal with the code Si2824Pr11 shows no significant distribution of the molar mass with respect to the origin of the different measured samples within the limits of uncertainty. The bias of single values can be clearly traced back to experimental scattering. The latter is more evident in this crystal material, because of the extreme enrichment in ^{28}Si (x(^{28}Si) > 0.999 99 mol/mol) which is associated with a respective reduced uncertainty (without the statistical contribution). The average relative combined uncertainty of the molar mass u_{c,rel}(M) = 2.7 × 10^{−9} legitimates the use of this new crystal material being implemented in the available very limited pool of enriched silicon spheres as primary reference standards.
Acknowledgments
The authors gratefully acknowledge discussions with Stefan Wundrack (PTB) and Nikolay Abrosimov (IKZ). Many thanks to Daniela Eppers (PTB) for providing the samples from distinct crystal locations.
Author contribution statement
Both authors contributed equally to the experiments and writing of this paper.
References
 Resolution 1 of the 26th CGPM (2018) On the revision of the International System of Units (SI), https://www.bipm.org/en/CGPM/db/26/1/ (accessed September 2019) [Google Scholar]
 Draft 9th edition of the SI brochure (2019) The International system of Units SI, https://www.bipm.org/utils/en/pdf/sirevisedbrochure/DraftSIBrochure2019.pdf (accessed September 2019) [Google Scholar]
 T.C. Liebisch, J. Stenger, J. Ullrich, Ann. Phys. 531, 1800339 (2018) [Google Scholar]
 D.B. Newell et al., Metrologia 55, L13 (2018) [Google Scholar]
 I.A. Robinson, S. Schlamminger, Metrologia 53, A46 (2016) [Google Scholar]
 R.D. Deslattes et al., Phys. Rev. Lett. 33, 463 (1974) [Google Scholar]
 P. Becker, H. Bettin, Philos. Trans. R. Soc. A 369, 3925 (2011) [CrossRef] [Google Scholar]
 K. Fujii et al., Metrologia 53, A19 (2016) [Google Scholar]
 A. Pramann, O. Rienitz, Anal. Chem. 88, 5963 (2016) [CrossRef] [PubMed] [Google Scholar]
 N. Abrosimov et al., Metrologia 54, 599 (2017) [Google Scholar]
 BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML 2008 Evaluation of measurement data − Guide to the expression of uncertainty in measurement JCGM 100:2008 [Google Scholar]
 A. Pramann, K.S. Lee, J. Noordmann, O. Rienitz, Metrologia 52, 800 (2015) [Google Scholar]
 A. Pramann, T. Narukawa, O. Rienitz, Metrologia 54, 738 (2017) [Google Scholar]
 O. Rienitz, A. Pramann, D. Schiel, Int. J. Mass Spectrom. 289, 47 (2010) [Google Scholar]
 A. Pramann, O. Rienitz, D. Schiel, B. Güttler, S. Valkiers, Int. J. Mass Spectrom. 305, 58 (2011) [Google Scholar]
 L. Yang, Z. Mester, R.E. Sturgeon, J. Meija, Anal. Chem. 84, 2321 (2012) [CrossRef] [PubMed] [Google Scholar]
 T. Narukawa, A. Hioki, N. Kuramoto, K. Fujii, Metrologia 51, 161 (2014) [Google Scholar]
 R.D. Vocke Jr., S.A. Rabb, G.C. Turk, Metrologia 51, 361 (2014) [Google Scholar]
 T. Ren, J. Wang, T. Zhou, H. Lu, Y.j. Zhou, J. Anal. At. Spectrom. 30, 2449 (2015) [Google Scholar]
 G. Mana, O. Rienitz, Int. J. Mass Spectrom. 291, 55 (2010) [Google Scholar]
 M. Wang, G. Audi, F.G. Kondev, W.J. Huang, S. Naimi, X. Xu, Chin. Phys. C 41, 030003 (2017) [NASA ADS] [CrossRef] [Google Scholar]
 A. Pramann, O. Rienitz, D. Schiel, B. Güttler, Int. J. Mass Spectrom. 299, 78 (2011) [Google Scholar]
 M.E. Wieser, J.B. Schwieters, Int. J. Mass. Spectrom. 242, 97 (2005) [Google Scholar]
 R. Kessel, M. Berglund, R. Wellum, Accred. Qual. Assur. 13, 293 (2008) [CrossRef] [Google Scholar]
 S.L.R. Ellison, A. Williams, EURACHEM/CITAC Guide CG4: Quantifying Uncertainty in Analytical Measurement (2012), https://www.eurachem.org/index.php/publications/guides/quam#translations [Google Scholar]
 BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML 2008 Evaluation of measurement data − Supplement 1 to the “Guide to the expression of uncertainty in measurement” − Propagation of the distributions using a Monte Carlo method, JCGM 101, 2008 [Google Scholar]
Cite this article as: Axel Pramann, Olaf Rienitz, The molar mass of a new enriched silicon crystal: maintaining the realization and dissemination of the kilogram and mole in the new SI, Eur. Phys. J. Appl. Phys. 88, 20904 (2019)
All Tables
Operating conditions of the MCICPMS with components almost siliconfree (PFA: perfluoroalkoxy alkane, PEEK: polyether ether ketone).
Exact positions of the silicon samples in the crystal and respective molar masses.
Arithmetic mean values of M, x(^{28}Si), x(^{29}Si), and x(^{30}Si) of the different available Si crystals highly enriched in ^{28}Si. Numbers in brackets denote uncertainties in the last digits (k = 1).
All Figures
Fig. 1 Photograph of the final single crystalline silicon crystal ingot Si2824Pr11 (length: 52 cm, mass: 5575 g) used for the spheres Si28kg02a and Si28kg02b. 

In the text 
Fig. 2 Final silicon crystal ingot (Si2824Pr11). Left: in the float zone apparatus at the Institute of Crystal Growth (IKZ), Berlin, Germany; right: main parts already cut (the large parts are reserved for the two spheres). 

In the text 
Fig. 3 Schematic cross section of the silicon crystal Si2824Pr11. The samples measured were taken from parts L, N, S, V, and W. The regions N, S, and V are bracketing the two spheres in the segments P and T. Each sample (cubic shape) has an approximate mass of 500 mg. 

In the text 
Fig. 4 Cross section of the disc of part V. In the lower centre region, five adjacent samples V1.1.1–V1.1.5 are indicated (compare Fig. 3). 

In the text 
Fig. 5 Inductivelycoupled plasma visible via a new sapphire bonnet type with reduced content of silicon with natural isotopic composition. 

In the text 
Fig. 6 Average values of molar masses of the 17 individual crystal samples of the new crystal Si2824Pr11 measured in the current study. Combined uncertainties (k = 1) are indicated by error bars. The arithmetic mean is M = 27.976 933 787(77) g/mol (indicated by the solid line). The relative combined uncertainty associated with the average molar mass is u_{c,rel}(M) = 2.7 × 10^{−9} (including the contribution of scattering). Upper and lower limits of this uncertainty are shown by red dashed lines. The dotted black lines indicate the upper and lower uncertainty limits without the contribution of scattering. 

In the text 
Fig. 7 Threedimensional molar mass plots showing the axial and radial origins of the samples together with the respective molar masses. The plots serve as a rough indication of the sample origin in the initial crystal ingot (Fig. 1). Detailed molar mass distributions are shown in Figure 6. 

In the text 
Fig. 8 Degrees of equivalence d_{i} of the results of the average values of the molar mass of the 17 crystal samples. Error bars denote the expanded uncertainties (k = 2) associated with the d_{i} . Note that data encompassing the zero line with their uncertainties are consistent with the average molar mass. This uncertainty still does not cover the scattering described in the text. 

In the text 
Fig. 9 Individual measurements (sequences) of the molar mass of the crystal sample Si2824Pr11L.1.1 with the individual combined uncertainty (k =1). The solid line represents the average molar mass of sample L.1.1 with upper and lower limits of the average uncertainty (dashed lines). 

In the text 
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